Jackson fired home an early opener before being felled in a collision with St Mirren goalkeeper Paul Gallacher midway through the first half.
His team-mates soon moved two ahead through Sam Parkin's brilliant header before Sean Lynch's deflected 36th-minute effort quickly gave St Mirren hope.
But Gareth Wardlaw, St Mirren's only available striker, missed a great chance late in the first half and the visitors failed to launch a second-half comeback.
St Johnstone were comfortable without being entirely convincing but deserved the three points as they leapfrogged the visitors to move off the bottom.
Both sides were seeking to make amends and Derek McInnes handed Jennison Myrie-Williams a debut as he made six changes. Allan Maybury and Jackson were recalled while Jody Morris, Peter Enckelman and Liam Craig returned from injury.
St Mirren brought in Nick Hegarty, Sean Lynch, Steven Robb and David van Zanten following their defeat at Kilmarnock but Steven Thomson was absent despite being expected to shake off an injury.
McInnes had told his players their performance in a 2-0 home defeat by Motherwell last weekend was unacceptable but they quickly set about atoning for that insipid display when Jackson struck in the third minute. He kept Danny Grainger's inswinging free-kick in and forced Paul Gallacher to block at his near post before getting up to squeeze the rebound high into the net.
The striker inadvertently blocked Craig's goalbound effort following a free-kick from a similar distance as St Johnstone continued to edge a quiet opening quarter.
Jackson's luck went from bad to worse midway through the half when Gallacher accidentally caught the striker after rushing off his line to punch away Dave Mackay's long ball.
Morris fired wide as his team-mate lay motionless and he was carried off on a stretcher after a six-minute delay, although he soon recovered in the dressing room despite a sore jaw.
Collin Samuel came on but his side almost lost their lead immediately when Lynch lobbed the ball into the six-yard box following a half-cleared corner, however Enckelman stopped Lee Mair's header right on his line.
St Johnstone soon went two ahead in the 33rd minute when Sam Parkin brilliantly headed home. Myrie-Williams exchanged passes with Maybury before sending in a driven cross that Parkin expertly guided into the far corner from 15 yards.
Myrie-Williams should have put the game beyond the Paisley Saints in the 36th minute when John Potter failed to deal with a long ball to leave the winger with a clear run on goal, but his left-footed effort allowed Gallacher to block with his foot.
The visitors took full advantage of the let-off within seconds when the ball fell for Lynch 30 yards out. He powerfully struck a first-time effort and saw it take a major deflection before nestling in the corner of the net.
Wardlaw missed a great chance to level in the 45th minute when the ball fell invitingly for the former Cowdenbeath striker six yards out following Hegarty's cross, but he sliced well wide.
St Johnstone calmed the match down after the break and Murray Davidson headed just over in the 55th minute after an instant cross from Maybury.
The home side remained in the ascendancy but their final pass or delivery from set-pieces let them down too often and they had no-one else in the box when Samuel pulled the ball across goal after Morris did cut open the visitors.
St Mirren struggled to get any attacking rhythm going and Danny Lennon moved Darren McGregor from defence into attack, but the former Cowdenbeath player headed over from 12 yards following Hegarty's injury-time corner.